Actual Infinity in Reality?

[polymath257]

Back to the topic.... I think there is an question that wasn't answered, that should be.

  If it is said, that a line contains a continuum of points (however you choose to define them).  Despite the fact, that this supposed infinity ends at 1 which is contradictory to saying that it is infinite in number in itself. (note:  I'll use one as a destination in this writing, although it may be another length)   What is the point immediately prior to 1?  There is necessarily an instance, where you transition from "not 1" to "1" while traveling along this line. 

I don't think that those proposing an actual infinite can answer the question.   I believe that this question shows the bait and switch that is occurring (whether the presenter knows it or not).  I think it is also why I have found difficulty in these conversations in having someone define what the term "point" is.  (It much easier to play fast and loose, if you do not define your terms).  If the points along the line are in fact infinite, then there cannot be a transition from "not 1" to "1".  As the argument goes, no matter how small the number is between our last point and the destination, we can always make up another number which is yet smaller (nature of the decimal system).  And we can repeat this over and over again, never reaching 1.  The time doesn't matter; this will never end (which is correctly  the definition of infinite) .   This is what Zeno's dichotomy (runners) paradox shows .   And I don't think that this is being addressed.  To get from "not 1" to "1"  you have to end the infinity (thus not infinite).

If you follow the logic and the procedure that is used to get an infinity in this way, then you cannot logically reach the destination either (not if you are consistent).  Adding an infinity of points of time, does not change here that the process will never end (which is why time is inconsequential).  The fact, that it does end, and that motion is possible, shows that this idea of a infinity in any given line and any given motion, shows that this idea is not logical (or at least the way it is argued is not logical).  

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Why do you assume there is a point just before 1? In fact, I can prove that, in fact, there is no point that is the 'last point before 1'.

The proof is a simple one done via proof by contradiction. Suppose x is any point before 1, so x<1. Let y be the average of x and 1. Then x<y<1, showing x is NOT the last point. hence, no last point can exist.

We cannot 'reach the destination' IN THAT WAY. We still reach the destination, but not via stopping at each of those infinitely many points.

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Ok... .so how would you answer?  Would you agree, that if there is a logical contradiction, that this would disprove the math?   Perhaps a problem in the assumptions or in the way things are being explained, that it is not an actual infinite but something else. 

This guy has a number of articles about Infinity and what he sees as philosophical problems underpinning this relatively recent theory.  The following observation is interesting. 

Mathematicians are an interesting bunch. They are very, very rigorous when it comes to analyzing implications – what follows from what. They do not seem nearly as rigorous when it comes to analyzing presuppositions – what precedes from what. In fact, they do not even seem to be aware of their own presuppositions. I’ve been told countless times, “It’s absolutely certain that Cantor proved the existence of different sizes of infinite sets! Mathematicians have double-checked his work for a century!”

But they don’t seem to be aware of one problem: what if the presuppositions of Cantor’s proof are wrong? What if – specifically – the concepts that he presupposed were imprecise.

What mathematicians have shown is that the axiom of infinity doesn't lead to any contradictions. The concepts are quite precise and the assumptions are very clearly laid out. You can look at the axioms for ZFC set theory if you wish.

Your reference is just wrong. Mathematicians extensively looked at the basic assumptions. That was part of the revolution in math about a century ago. Many alternatives were explored and debated. The current axioms were the result of those discussions.

Again, truthfully, the distinction between actual and potential infinities is obsolete. It really hasn't been a part of serious mathematical discussion for well over a century. The only people still mentioning it are Aristotelian philosophers. And, truthfully, their ideas are also mostly out of date. Even their logic is obsolete and has been replaced by propositional and predicate logic.

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So from Hilbert's Hote we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

These are contradictory statements resulting from simple arithmetic operations (from 2). 

These are NOT contradictory. The last one (infinity-infinity=3) is nonsense, but you derived it incorrectly. It is in the same category as 0/0=3 following from 0*3=0.

The rest are actually true.

What do you think is the contradiction? Be clear.